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1 The Tenuous Trade-Off between Risk and Incentives Author(s): Canice Prendergast Source: The Journal of Political Economy, Vol. 110, No. 5 (Oct., 2002), pp Published by: The University of Chicago Press Stable URL: Accessed: 13/07/ :41 Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at JSTOR's Terms and Conditions of Use provides, in part, that unless you have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you may use content in the JSTOR archive only for your personal, non-commercial use. Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained at Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission. JSTOR is a not-for-profit organization founded in 1995 to build trusted digital archives for scholarship. We work with the scholarly community to preserve their work and the materials they rely upon, and to build a common research platform that promotes the discovery and use of these resources. For more information about JSTOR, please contact The University of Chicago Press is collaborating with JSTOR to digitize, preserve and extend access to The Journal of Political Economy.

2 The Tenuous Trade-off between Risk and Incentives Canice Prendergast University of Chicago and National Bureau of Economic Research Empirical work testing for a negative trade-off between risk and incentives has not had much success: the data suggest a positive relationship between measures of uncertainty and incentives rather than the posited negative trade-off. I argue that the existing literature fails to account for an important effect of uncertainty on incentives through the allocation of responsibility to employees. When workers operate in certain settings, firms are content to assign tasks to workers and monitor their inputs. By contrast, when the situation is more uncertain, they delegate responsibility to workers but, to constrain their discretion, base compensation on observed output. I. Introduction Much of the empirical and theoretical work on agency issues concerns the trade-off of risk and incentives. From this perspective, the cost of offering a pay-for-performance contract to a (risk-averse) employee is that it imposes risk on his compensation, which causes higher wage costs. Consequently, when choosing higher performance pay, firms trade off the benefits of more effort against higher wage costs. The risk imposed on workers is increasing in the uncertainty of the environment so that the standard test of the trade-off is to show that incentive pay is lower in more uncertain environments. Unfortunately, empirical re- Thanks to Jim Baron, Wouter Dessein, Luis Garicano, Bob Gibbons, Tom Hubbard, Sendhil Mullainathan, Tano Santos, Michael Raith, Jan Zabojnik, an anonymous referee, the editor, and participants at the University of Chicago, NBER, and the American Compensation Association conference, for helpful comments. This work was supported by the National Science Foundation. Any errors are my own. [ournal of Political Economy, 2002, vol. 110, no. 5]? 2002 by The University of Chicago. All rights reserved /2002/ $

3 1072 JOURNAL OF POLITICAL ECONOMY search has not shown a convincing relationship between pay for performance and observed measures of uncertainty. Indeed, as described below, for a range of occupations the data suggest that observed measures of uncertainty are positively correlated with incentive provision. The purpose of this paper is to understand why this is the case and to offer an alternative theory of how uncertainty affects incentive provision. In agency models, the uncertainty of the environment typically has one effect: it adds observation error to performance measures. This leads to the negative relationship between uncertainty and incentives. This paper argues that there is another effect of uncertainty on incentive provision that may be more important, namely, the delegation of responsibility to employees. The paper focuses on the distinction between instances in which an employer tells his agent what to work on and situations in which the agent is given discretion over the activities that he spends time on. The results of the paper are based on two implications of this choice. First, delegation is more likely when there is greater uncertainty about what the agent should be doing. Second, output-based incentive pay is more likely to be observed in cases in which employees have considerable discretion: there is little need to base pay on output when inputs are monitored. So uncertain environments result in the delegation of responsibilities, which in turn generates incentive pay based on output. Thus uncertainty and incentives are positively correlated. The idea is best described by the following example. A firm is involved in large-scale construction projects around the world and uses project managers to run those projects. Compare projects being carried out in Canada to those in, say, Armenia. The company has considerable experience in Canada and "knows the ropes" for doing business there. By contrast, the company is very much in the dark when doing projects in Armenia, both because the economic environment is so different from what it is accustomed to and because it has little experience there. If one were to make predictions about compensation using the standard trade-off between risk and incentives, one would expect to see more pay for performance for the project manager in Canada than in Armenia since the manager's performance can be measured more precisely in Canada. This is exactly the opposite of what happens: in fact, the Canadian manager is paid a salary (with a small bonus), whereas the Armenian manager's pay is tied to the profitability of the project. The reasoning is simple. The company believes that since it has a good idea of how business should be done in Canada, it takes control of most decision making and monitors the manager largely on the basis of his inputs. Headquarters feels that it can make effective decisions in Canada and therefore does not delegate much decision-making power to the manager. By contrast, headquarters has little ability to determine the

4 RISK AND INCENTIVES 1073 profitability of the Armenian projects, since this depends on many pieces of information that it does not have. Because it believes itself to have such a poor handle on the business, it chooses an alternative strategy: it (largely) delegates decision making to the project manager. But it also offers him an output-based contract since this is the only way to monitor his performance. In effect, output-based pay is used because in uncertain environments, there are no other good measures by which to align incentives. Thus incentive pay and uncertainty are positively correlated, in contrast to the standard model of trading off risk and incentives. The modeling sections of the paper are largely devoted to outlining this effect, discussing its implications, and analyzing the robustness of its conclusions. I begin in Section II by describing the existing empirical evidence on the trade-off of risk and incentives. Evidence on the trade-off comes from three areas: (i) executives, (ii) sharecroppers, and (iii) franchisees. First, the evidence on executive compensation about whether risk and incentives are substitutes is mixed. Some studies find evidence in favor of the negative trade-off, others find evidence of a positive relationship, and some find no relationship. Second, the evidence on agricultural sharecropping clearly points to a positive relationship between measures of risk and the fraction of output received by sharecroppers. Third, firms often decide whether to franchise their retail outlets or to retain them as company-owned. For franchisees, strong output-based contracts are the norm, whereas in company-owned stores, pay for performance is more muted. Contrary to the trade-off of risk and incentives, again there is a clear positive relationship between measures of risk and the decision to franchise: thus pay for performance is more common in risky settings.1 Most work in agency theory assumes that the tasks carried out by employees are fixed and then considers the optimal output-based contracts given that restriction. At one extreme of this spectrum, workers are residual claimants on output, whereas at the other they are offered few incentives without output-based pay. But realistically, when outputbased pay disappears, firms do not simply resign themselves to workers' exerting little effort. Instead, they find other means of monitoring, namely by assigning the agent to carry out certain tasks and by direct 1 Beyond these systematic studies, the theory also seems a little strained at a more anecdotal level in that the theory suggests that pay for performance would be more likely to be observed in stable industries and time periods in which there is little extraneous risk on workers. While I know of no recent systematic work on the cross-industry nature of pay for performance, it appears that incentive pay is used mostly in volatile industries, such as the use of options in high-tech industries and bonuses in the financial sector. Lazear (2000) also makes this point. If the trade-off of risk and incentives is the primary force determining pay for performance, these are hardly the candidates that would be predicted to be most likely to use such risky instruments.

5 1074 JOURNAL OF POLITICAL ECONOMY observation of the agent's inputs. This in itself is not a problem for the existing theory unless the marginal cost of using this combination of directed actions and input monitoring depends on the uncertainty in the environment. But I argue that there is a natural relationship between the effectiveness of directed action and uncertainty. In particular, in stable scenarios, a principal has a good idea of what the agent should be doing, so that by observing efforts, he can be pretty sure that private and social benefits are aligned. However, in less certain environments, the principal may be able to monitor inputs (e.g., whether the agent is keeping busy) but is likely to have less idea about what the agent should be spending his time on. In the absence of an effective mechanism for revealing this information, the principal is likely to respond by offering a pay-for-performance contract. In other words, input monitoring will be used in stable settings, but less so in more uncertain environments, where workers will be offered more discretion but will have their actions constrained by tying pay to performance. The reason for this result is simply that the expected marginal return to using output-based contracts to align socially and privately optimal actions is greater in uncertain situations than in stable ones, so that incentive pay is positively related to uncertainty. Obvious though this point may be, it remains unobserved in the literature, which has typically assumed that the marginal return to actions is independent of the underlying riskiness of the environment. I argue that this assumption is not plausible in many settings, and introducing a relationship between uncertainty and the marginal return to agent actions explains why we see more pay for performance in rapidly changing industries such as the high-tech sector,2 whereas in more stable settings, input-based contracts will be the norm. The basic insight of the model is described in Section III, where the optimal trade-off between input monitoring and output monitoring is shown to depend on the underlying uncertainty of the environment. Section IV considers further applications and extensions of the model. First, I argue that in contrast to the implications of the existing literature, this paper suggests that pay-for-performance contracts are more likely to be found in complex than in straightforward jobs. The reason for this is that it is more difficult to monitor in complex positions, since the optimal action is hard to pinpoint, compared to jobs in which there is little doubt over the right course of action. Hence complexity and incentive pay go hand-in-hand. Second, I consider a series of modeling extensions in which I show that the insights of the model are robust to such extensions as allowing communication between the principal and 2 One needs to be a little careful here to distinguish these firms from start-ups, which may offer options and stocks rather than high salaries for reasons of liquidity constraints.

6 RISK AND INCENTIVES 1075 agent and to cases in which partial delegation of responsibilities may be optimally used. Finally, I argue that the basic results are also likely to hold with other forms of uncertainty. In Section V, I show that one critical assumption generates the positive trade-off between uncertainty and incentives: the availability of good measures of output. It is well known that when output measures are unreliable, the desirability of output-based contracting falls because of multitasking concerns. This in itself is not a problem for the theory; what does matter is how multitasking concerns vary with the uncertainty of the environment. Specifically, if multitasking concerns are greater in uncertain environments, then the results described above can reverse in that the standard negative trade-off can now be attained. To see why, consider the example of the construction company in Armenia and Canada. An obvious multitasking problem that can arise with construction projects is that the accounts can be "doctored" to show that a project is doing well when in fact it is a bust. This doctoring is done by the project manager to avoid canceling an inefficient project from which the manager gains private benefits. If the extent to which accounts can be manipulated does not vary between Canada and Armenia, the qualitative results of the basic model are unchanged: multitasking makes output-based contracting less desirable, but there are no qualitative implications for how uncertainty affects the trade-off between input and output monitoring. But if it is easier to distort performance measures in more uncertain environments (e.g., since accounting methods in Armenia are nonstandard), then the results can reverse in that a negative trade-off is now possible. The reason for this reversal is that there is now a countervailing effect whereby output-based contracts become increasingly distorted in uncertain settings. If the distortion increases suf- ficiently rapidly with uncertainty, it can swamp the beneficial returns to delegating. The upshot of this is that a positive trade-off of risk and incentives should be predicted only in situations in which good measures of performance are available, such as sharecropping and franchising, rather than in occupations in which observed measures are a poor reflection of performance. The paper is largely motivated by the absence of a negative trade-off between incentives and observed measures of uncertainty in the empirical literature. I return to the empirical evidence in Section VI, where I consider the implications of this model for observed outcomes. I argue that the observed evidence is better explained by this model than the standard agency model. In standard econometric parlance, a difficulty with the existing empirical work is omitted variable bias. This model argues that uncertainty affects the responsibilities offered to workers, which in turn affects incentives. But responsibilities and discretion are rarely observed by the econometrician, so that omitted variable bias

7 1076 JOURNAL OF POLITICAL ECONOMY arises. I use existing evidence from the literature on franchising and sharecropping to address this and to argue that the data appear consistent with the insights of the model. In the most general incarnation of the standard agency model, such as Grossman and Hart (1983), it is difficult to generate simple agency contracts in which one can talk about more or less incentives. The benchmark here is not such a general model, but the more commonly used model offered by Holmstrom and Milgrom (1991), where linear contracts are generated by assuming a dynamic model with exponential utility and normally distributed errors. For our purposes, this benchmark has two attractive features. First, optimal contracts are linear in measures of output, and second, the (sole) measure of incentives is decreasing in the (sole) measure of risk for a normal distribution, the variance. Anyone familiar with recent contributions to agency theory will recognize the by now common optimal piece rate 1/[1 + rc"(e)a2], where ris the degree of absolute risk aversion, C(e) is the cost of effort to the agent, and a2 is the variance of the measurement error on performance. This yields the simple prediction that the piece rate will fall with the variance. I construct a model that in spirit is similar to this, though with risk-neutral agents, and show how the optimal piece rate increases with the measure of uncertainty, a2. It is also worth emphasizing at the outset that I make no claim that a positive relationship between observed measures of risk and incentive provision should necessarily be observed. Uncertainty surely affects incentive provision through the risk costs that are the focus of the existing literature. Instead, my claim is simply that there are plausible influences that can cause a positive relationship and they may be important enough to dominate. I begin in Section II by documenting empirical evidence on the relationship between measures of risk and the incentives provided to workers, and I illustrate the paucity of evidence on a negative relationship between these variables. Following this, Section III considers the tradeoff between directing the actions of employees (and monitoring inputs) and delegating the choice to the worker (with endogenously optimal output-based monitoring). I show that the cost of assigning tasks to the agent is increasing in the uncertainty of the environment so that firms prefer to use output monitoring if the environment is risky enough. Section IV considers some implications and extensions of the model, and Section V considers the importance of good output measures. I examine the empirical implications of the model in Section VI, and I conclude the paper with a brief discussion in Section VII.

8 RISK AND INCENTIVES 1077 TABLE 1 THE TRADE-OFF OF RISK AND INCENTIVES FOR EXECUTIVES Authors Measure of Risk Result Lambert and Larcker (1987) Volatility of returns Garen (1994) Volatility of returns 0 Yermack (1995) (options only) Variance of returns 0 Bushman et al. (1996) Volatility of returns 0 Ittner et al. (1997) (full sample) Correlation of finan- 0 cial and accounting returns Aggarwal and Samwick (1999) Volatility of returns Core and Guay (1999) Idiosyncratic risk + Conyon and Murphy (1999) Volatility of returns 0 Jin (2000) Idiosyncratic risk Core and Guay (in press) Volatility of returns + Oyer and Shaefer (2001) Options grants + I. The Evidence This section provides a backdrop by surveying the existing evidence on how pay for performance varies with uncertainty. In each of the studies below, I use the following characterization: a minus sign implies that there is statistically significant evidence (at the 5 percent level) of a negative relationship, as predicted by the theory; a plus sign implies a significant positive relationship; and a zero means that there is no statistically significant relationship. For reasons that will become clearer below, I consider four different classes of occupations in which the tradeoff has been tested: (i) executives, (ii) sharecroppers, (iii) franchisees, and (iv) sales force workers and others. Table 1 gives the evidence on chief executive officers (and sometimes other executives), providing both the measure used and its results.3 Here the evidence is inconclusive, with three studies finding a statistically 3 In each of these cases, the test carried out addresses how the fraction of the firm held by the executive varies with some measure of uncertainty. The theory addresses how "piece rates" vary with uncertainty, so this is the closest available measure. Several studies present many sets of results, and I provide the most comprehensive model. Some elaboration is needed for two cases here. First, Ittner, Larcker, and Rajan (1997) offer some results with a significant negative effect and others with no significance. In a simple ordinary least squares regression, there is a negative relationship, but when endogeneity is controlled for via a structural estimation technique for their full sample, no noise variable is significant at the 5 percent level. I cite this latter result here. Second, Core and Guay (1999) offer a series of results, mostly on total equity compensation rather than on the share owned by the executive. The results cited here are taken from table 7 in their paper, which carries out the desired regression with %ownership as the dependent variable and shows a positive relationship with idiosyncratic risk.

9 1078 JOURNAL OF POLITICAL ECONOMY TABLE 2 THE TRADE-OFF OF RISK AND INCENTIVES FOR SHARECROPPERS Authors Measure of Risk Result Rao (1971) Variance of profits + Allen and Lueck (1992) Coefficient of variation in + Allen and Lueck (1995) yield Coefficient of variation in + yield (within crop) significant negative effect, three a significant positive relationship,4 and five others finding no relationship between risk and incentives. Table 1 offers weak evidence about the extent to which incentives are traded off against risk. While there may be such a trade-off, it hardly jumps out in the data.5 At the very least, this suggests a role for considering other potential explanations for the relationship between uncertainty and contracts. Yet executives appear to be the occupation with the strongest evidence in favor of the trade-off. To see this, I now consider two other occupations that have attracted some testing of this hypothesis, namely, sharecropping and franchising. The evidence on sharecropping in table 2 shows that the fraction of output sharecroppers keep is increasing in the noisiness of the financial returns, the opposite of the outcome suggested by the theory.6 Equally, consider the relationship between the decision to franchise (with high pay for performance) and the decision to keep a store company-owned (with less pay for performance).7 If the traditional trade-off is correct, we should expect to see a negative relationship between franchising and riskiness. However, table 3, taken from Lafontaine and Slade (2001), suggests the opposite. As Lafontaine and Slade (2001) conclude, "these results suggest a robust pattern that is unsupportive of the standard agency model" (p. 10). Tables 2 and 3 should, I believe, be seen in combination as strong 4 Oyer and Shaefer (2001) consider the use of option grants that are more broadly based than simply to executives. As such, this paper may be misplaced by including it with other studies on executives. However, the form of compensation (options) is so closely aligned to those of the other studies that I decided to include it here. 5 It is also worth pointing out here that some of these studies consider only the cash compensation and bonus of executives (e.g., Lambert and Larcker 1987; Bushman, Indjejikian, and Smith 1996; Ittner et al. 1997), which is only a component of total incentives. However, restricting attention to only those papers that also include stock- and optionbased compensation does not provide any more conclusive results. 6Ackerberg and Botticini (2002) use a different strategy to test for risk sharing in sharecropping by considering how farmer wealth, a proxy for ability to handle risk, affects contract choice. They find that after they control for matching issues, more wealth is correlated with a greater likelihood of renting, consistent with the usual risk-sharing story. 7 Obviously, this raises the question of why asset ownership affects incentives. See Holmstr6m and Milgrom (1991) on this.

10 RISK AND INCENTIVES 1079 TABLE 3 RISK AND THE DECISION TO FRANCHISE Authors Industry Result Martin (1988) Panel across sectors + Norton (1988) Restaurants and hotels + Lafontaine (1992) Many sectors (business for- + mat franchising) Anderson and Schmittlein (1984) Electronics components 0 John and Weitz (1988) Industrial firms 0 evidence against the traditional negative trade-off. The most generous interpretation of these data is that there is little evidence in its favor (particularly when one factors in the possibility of publication bias). Indeed, there is more evidence of a positive relationship between uncertainty and incentive provision, particularly outside the market for executives. It is this issue I explore in the theoretical sections of the paper.8 Before doing so, for the sake of completeness, I include other tests on the trade-off in table 4 for a range of occupations, which show little systematic pattern. II. Delegation and the Choice between Monitoring Inputs and Outputs Firms do not choose compensation plans independent of other strategic decisions. In this section, I consider how the decision to delegate decision-making power affects contracts offered to workers and how that decision depends on the uncertainty of the economic environment. Specifically, firms delegate decision-making power more in uncertain environments but offer output-based contracts in order to constrain the possibility that they use their discretion in harmful ways. By contrast, in more certain environments, firms assign tasks to workers and find it more profitable to monitor actions directly. This section formalizes this in the following way. A principal hires an agent to exert effort on one of n possible tasks. For the action on which the agent is employed, i, he chooses an effort level to exert, ei, where 8 There are at least two reasons why we might find little relationship between observed measures of risk and incentive pay. First, it may be that the empirical measures of risk and pay for performance are a poor reflection of the true environment facing employees. This is the standard empirical measurement error problem, and I have little to say about it. This is less likely to be a problem for the literature on executives or franchisees, since we observe the contracts they receive and can relatively easily compute the relevant measures of risk. For studies that address employees on implicit contracts, as in Brown (1990) or MacLeod and Parent (1999), this problem may be more severe. Second, it could be that our theories are missing something important about the relationship between the desire to induce individuals to exert effort and the riskiness of the environments in which they find themselves. It is this second point that I address here.

11 10o80 JOURNAL OF POLITICAL ECONOMY TABLE 4 RISK AND INCENTIVES IN OTHER INDUSTRIES Authors Industry Result Kawasaki and McMillan (1987) Japanese subcontractors Leffler and Rucker (1991) Timber tracts + Mulherin (1986) Natural gas contracts + MacLeod and Parent (1999) Many sectors 0 Coughlin and Narasimhan (1992) Sales force workers 0 John and Weitz (1989) Sales force workers 0 Hallagan (1978) California gold mining 0 i refers to activity i. Output from exerting effort on task i depends on the effort level and on a random variable Pi. Specifically, the performance of the firm is given by Yi = Pi + ei. The cost of effort on activity i is C(ei), which has the following standard properties: C'(e) > 0, C"(ei) > 0, and C'(O) = 0. The distribution of the n random variables Pi is given by (i. The distributions differ only in their means: they are distributed according to a common distribution with mean Pi and variance a2. In what follows, I vary a2; an increase in a2 identifies a more uncertain environment. This is the measure of risk considered throughout the paper. All individuals are risk-neutral, and throughout the paper, the reservation utility of the agent is normalized to zero. I model the agents as risk-neutral in order to ignore the standard trade-off. The premise of this section is that agents often have information not available to the principal.9 I assume that the agent knows the true values of pi for all i, whereas the principal knows only the distribution of the Pi. I also assume that the technology is such that the agent can carry out only a single activity; that is, the fixed cost of engaging in two activities is too large to make it worthwhile to get involved in more than one. The principal can potentially collect two pieces of information with which to reward the agent. First, she can observe the efforts exerted by the agent, ei, at a monitoring cost me. Second, she can collect information on output produced by the agent. This costs my to collect. Throughout this section, I assume that my > me. The monitoring costs of output are a metaphor to reflect any costs of introducing a pay-for-performance plan, such as multitasking concerns. (See Sec. IV for details.) If the agent is indifferent about which activity he carries out, the problem has the following trivial solution, under the assumption that it is at least profitable to monitor inputs. First, as me < my, the principal 9 For instance, an agent may know the right customers to focus on, which subordinate to let go, or, more generally, the right strategies that should be followed.

12 RISK AND INCENTIVES 1081 monitors inputs and offers a contract to the worker w(ei) = C(ei) for all i. In other words, simply offer the worker a contract that pays him his costs of effort. As the agent knows the true value of pi, he will choose the activity that yields the highest value of output among the n realized and will exert optimal effort on that activity. This optimal level of effort on that activity is given by e*, where C'(e*) = 1 (the isubscript is dropped for simplicity). This yields the first-best allocation of effort and activity selection and dominates any output-based contract, as me < my. But I assume that the agents have personal preferences over which activity they enjoy most (the activities have a personal benefit given by Bi). To keep matters simple, I assume that these benefits are small enough so that they do not affect the calculation of the optimal activity to carry out.10 The principal can, of course, extract these expected benefits through the salary paid to the agent.1 As in Prendergast and Topel (1996) and Aghion and Tirole (1997), I also assume that the principal does not know the preferences of the agent, that is, which activities have which benefits. The agent knows his private benefits of the various activities. As a concrete example, I consider the case in which the agents have personal preferences such that they are indifferent to n - 1 of the activities (Bi = 0) but gain a small benefit B> 0 from the last one; the principal has no idea of the identity of i. The principal believes the distribution over the preferred activity to be uniform. I assume that there is no correlation between Bi and Pi, though Section IV relaxes this assumption. Assigned actions and input-based contracts.-the principal has two functions: (i) to assign an allowable set of tasks that the worker can carry out and (ii) to decide how to reward the agent for the allowed set of tasks. First, consider a contract that rewards solely on effort (an inputbased contract) and leaves the choice of tasks entirely to the agent. If the principal offers the contract w(ei) = C(e,) for all i, the agent will simply carry out the activity that he enjoys most, since he gets rents of B from that activity. This will in expectation yield a surplus (and hence benefits to the principal) of 2= 1Pi + e* - C(e*) + B- me, n where 1 = C'(e*). If pi * pj for some i and j and B is small, this is dom- '0 If the benefits were large, the principal could allow the agent to carry out an activity even when it is output-dominated by another and charge the agent for carrying out that activity. 1 I ignore the agent's individual rationality constraint here by assuming that the worker signs a contract before observing pi. This allows up-front transfers such that the usual mechanism design issues with an informed agent can be ignored; instead, the principal's objective is to maximize surplus.

13 1082 JOURNAL OF POLITICAL ECONOMY inated by a strategy in which the principal restricts the allowed activities and offers w(ei) = C(e,) only for those activities. (If the agent carries out any other activities, he is simply penalized a sufficiently large amount of money that he will never do that activity.) Let activity k be the task with the highest mean, p,. If this is unique,12 the expected profits for the firm from this strategy are B Pk + e - C(e*) me. n As the private benefits are assumed small, this dominates delegating the activity choice to the agent if pi : pj, for some i and j. If the firm could monitor only inputs, this would be the optimal solution.'3 Note also that with assigned tasks, there is no value to offering an output-based contract since me< my. Delegated actions and output-based contracts.-now consider another option, to delegate the choice of action to the agent. This can be optimal only if the agent is paid on output; otherwise, the agent chooses the action with the highest private benefits. If output contracting is used, the optimal piece rate is such that the agent chooses the optimal level of effort, e*, and chooses the correct activity, j, that is, the one that maximizes i - C(e,) + Bi. In other words, the purpose of offering pay for performance here is not simply to induce effort, since an input-based contract could do this, but to induce the agent to carry out the right kinds of efforts. (This, of course, is nothing more than a relabeling of what effort means, but here the marginal return to inducing this kind of effort varies with the riskiness of the environment.) In order to determine the return to offering output-based contracts, consider the distribution of the first order statistic of the realization of the Pi. The reason this is necessary is that if the agent is offered an output-based contract, he will choose the highest realization of Pi among 12 In the case in which m of the n observations are tied with the highest mean pk, the optimal input contract allows the agent to work on any of these m tasks, and the profits from this strategy are mb Pk + e* - C(e*) m, n In the case in which all Pi have identical means, the agent is allowed to work on any activity. 13 I have restricted attention to the case in which w(ei) = C(e). With input monitoring, there is no better contract, though there are obviously others that can replicate the efficient outcome. In another possible mechanism, the principal auctions off the right to carry out various activities; e.g., the principal could offer a price at which the agent could carry out activity i. A natural case would be to offer the agent the opportunity to carry out activity k at no price but a positive price of B to carry out any other activity. But this cannot do any better than a contract that simply mandates the agent to carry out a task, since the agent is ex ante indifferent to all pi in an input monitoring plan.

14 RISK AND INCENTIVES the n. By contrast, if the agent is offered an input-based contract, he is assigned to activity k, which has mean P*. What matters then is the difference between the distribution of the first order statistic p' from the n realizations relative to pk. The idea here is that this difference is likely to be increasing in the variance of the environment. In other words, when the variance of the distribution of the pi is large, the value of sampling the first order statistic is larger than when it is small. If the variance is very small, the principal knows that if he simply assigns a task to the agent, the expected marginal cost of being mistaken is likely to be small. As a result, there is little cost to input monitoring. On the other hand, when a is large, the firm will likely use output-based monitoring, since there is little certainty about the right kinds of activities to engage in. I illustrate this with two cases in which I can get simple closed-form solutions: the normal distribution and the uniform distribution. There is one important difference between these two examples, namely between the case in which all actions look ex ante identical (example 1) and the case in which some actions look better than others from an ex ante perspective (example 2). Example 1: the normal distribution.-assume that n = 2 and that both random variables pi - V(O, a2). Therefore, all activities look identical to the principal. If input monitoring is used, the principal allows the agent to work on any activity and the expected surplus from input monitoring is e* - C(e*) - me + B. With output monitoring, the agent chooses the activity that maximizes surplus since a piece rate of one is optimal. As B is small, this implies that the agent chooses the activity with the highest Pi. The expected value of the first order statistic of the two random variables E[p'121] is E[p'f21] = a/r. Note that the first order statistic is increasing in a2 and that the profits from output monitoring are a B - + e* - C(e*) my, V7 n 1083 also increasing in a2. Output contracting is therefore preferred only if if and a B(n - 1) -> my-m+ () VTT n (if the right-hand side is nonnegative, as is necessarily the case when my > me). The agent is more likely to be offered an output-based contract

15 1084 JOURNAL OF POLITICAL ECONOMY in riskier environments since the critical level of variance, a2*, above which output monitoring is used is given by B(n- 1)' 2* = my- me+ r. (2) This trivial example provides the intuition for the paper's main results. When the environment is more uncertain, the cost of assigning the agent to carry out a particular action is high, since there is likely to be another with a significantly better return. As a result, the firm optimally chooses to delegate choice of action in sufficiently uncertain settings but constrains the agent's choice in that setting by basing pay on output. Thus incentive pay goes hand-in-hand with uncertainty. A more general result.-an important part of example 1 is that the returns to all the actions are drawn from the same distribution. For this case, a more general result arises. Assume that all the random variables are drawn from the same distribution and that yi = Pi + ei. In that case, there is a single critical value of a above which output contracts are optimal and below which assigned actions with input contracts are optimal. This arises from the fact that for a distribution Pi with common mean p and variance a2, the expected value of the first order statistic from n draws of a probability density function of the form f-lg[(pi - p)/a] can be characterized as E[plnj,] = p +,Hn, where H, is independent of a2 and p but depends on n. See Cox and Hinkley (1990) for details. Thus, as in example 1, the value of the first order statistic is linearly increasing in a, whereas the return to input contracts is not. Therefore, one can easily generalize the insights from example 1 to other distributions. A difficulty in proving general results arises for the case in which the random variables are drawn from distributions with different means. Problems arise in finding closed-form solutions to carry out comparative statics. For this case, I consider the simplest distribution in which closedform solutions are possible, namely, the uniform distribution, where we find results similar to those above. Example 2: the uniform distribution.-assume that pi U[-x + i, x + pi] and that n = 2. By renormalization, let the distributions be p, - U[-x, x] and p2 ~ U[-x + A, x + A], where A = p - P. Therefore, the two activities are uniformly distributed with common variance x2/3, but activity 2 has a mean that is A higher than activity 1. With input monitoring, the agent will be assigned to work on activity 2 and paid his costs of effort. This has expected return A + e* - C(e*) - me + (B/2). If offered an output-based compensation plan, the agent chooses the high-

16 RISK AND INCENTIVES 1085 est value of Pi, and expected profits are E[p',12] + e* - C(e*) + (B/2) - my. Then calculations similar to those in example 1 yield E[p'] Then delegation is preferred if -1-2 (2x- A) if A< 2x 6\ 2x/ - A = (3) 0 otherwise. 1 A \2 max 0, (2x- A) > my- m (4) But max 0, [1- (A/2x)]2(2x- A)} is nondecreasing in x (and nonincreasing in A), and the variance of the uniform distribution, x2/3, is increasing in x. Therefore, as the variance increases, so also (weakly) does the return to using an output-based contract to induce the agent to choose the activity correctly. Thus, once again, this example points to the positive correlation between risk and incentive pay based on outputs.14 To summarize, consider a more concrete example, the franchise decision. There is more use of pay for performance for franchisees than for managers of company-owned stores. Furthermore, we saw from table 4 that franchises are more common in uncertain environments than in certain settings. The interpretation that I place on this is that in uncertain environments, headquarters has less idea of the kinds of products that should be offered, their prices, the number of employees to hire, and so on than in more certain situations. As a result, it responds by offering output-based contracts, which are less necessary than when the headquarters knows with more certainty what should be done. At a more general level, this section raises what I feel is an aspect of the agency literature that is often overlooked, namely, that uncertainty is likely to affect both the compensation and optimal distribution of actions of employees. Typically in agency theory, we treat uncertainty in the economic environment as synonymous with measurement error; yet, as illustrated here, uncertainty has other effects on the employment relationship that confound the usual negative trade-off between risk and incentives. 14 It is important to note that I am not claiming that accuracy of monitoring is irrelevant to contracting problems. In fact, one can easily reinterpret these results to say that in more certain environments some measure of total incentives is at least as high as in less certain environments; all that differs is that monitoring occurs on inputs in the certain environments and on outputs in the less certain environments. But note that empirical researchers never see inputs, so that the objective of this paper has been to understand why empirical work that traces the relationship between outputs and uncertainty is unlikely to find a negative relationship.

17 o86 IV. Extensions JOURNAL OF POLITICAL ECONOMY In this section, I consider some implications of the model and also show that the basic insights are robust to other modeling assumptions. A. Complexity Recent contributions to agency theory focused on multitasking (Holmstrom and Milgrom 1991) suggest that complex jobs are less likely to use incentive pay. Complexjobs have many dimensions of performance, some of which are poorly observed, so that rewarding on the observed dimensions typically has harmful effects on the unobserved dimensions. Thus the marginal cost of output contracting in complex positions is high. However, it is also the case that the benefits of contracting on output are likely to be especially high in complex positions. Indeed, a simple parameterization of this model suggests a positive relationship between complexity and the likelihood of incentive pay. The reason for this is that it is more difficult to monitor complex positions than those for which it is easy to identify the optimal course of action; as a result, output-based contracts are more desirable in complex positions. A nat- ural measure of complexity in the model is given by n, the number of activities that the individual can carry out. In this model, as the number of activities increases, so also does the desire to induce the agent to choose the right one to work on. As a result, output-based pay is more likely in complex settings. To see this, consider example 1 with the normal distribution. Now suppose that the agent can choose among three activities, n = 3 rather than n = 2 in the basic setup. Let E[pllnt] be the expected value of the first order statistic from n draws. Then it is the case that 3a a = E[p(31} ] > E[P12=], = and output monitoring is used when a2 < a2*, where 2* 4r[m - me + (2B/3)]2 :2* (5) 9 Remember that the critical variance when n = 2 is given by 2*= / B\2 my,- me + 2r > a for B low, as is assumed here. Therefore, when the number of activities increases, the range of (variance) parameters in which output-based contracts are used also increases. What this suggests is that the returns

18 RISK AND INCENTIVES 1087 to offering incentive-based pay are greatest for complex jobs, where there is an overall measure of performance. The logic of this section is simply that those employees who carry out well-defined jobs, where the activities that keep the person (optimally) busy from one end of the day to the other are known, can easily be rewarded on input-based contracts. However, when the range of activities that the person engages in increases, it becomes harder to identify what the person should be doing, and so output-based contracts become necessary. B. Partial Delegation So far, I have described two options available to the principal: he either assigns a task to the agent or allows the agent unrestricted choice over actions. However, in some settings, it may be optimal to allow the agent to choose actions from a limited set, but a set that is not a singleton. This is not an issue when there are only two actions (as in examples 1 and 2). However, when there are more than two actions, the principal may partially delegate tasks in the following way: (i) for low levels of uncertainty, the principal assigns an action and monitors inputs (as above); (ii) for high levels of uncertainty, the principal allows the agent unrestricted choice over actions but with an output-based contract (again, as above); but (iii) for intermediate levels of uncertainty, he allows the agent to choose between a subset of actions and uses input monitoring. In effect, the principal excludes some actions, those that are believed to be the least profitable. As a result, allocation of tasks can vary more continuously with our measure of uncertainty (variance) than in the basic model above. Yet it remains the case that output contracting is used only in sufficiently uncertain settings. I illustrate this by another simple example in which n = 3 and the distributions are uniform but have different means. Example 3: the uniform distribution and n = 3.-Assume that there are three activities, 1, 2, and 3, where action 1 has the lowest mean payoff and action 3 has the highest. Specifically, let pi U[-x + (i- 2)A, x + (i - 2)A] and y, = Pi + ei. Therefore, the three activities are uniformly distributed with common variance x2/3, but activity 2 has a mean that is A higher than activity 1, and activity 3 has a mean that is A higher than activity 2. Three optimal outcomes are possible. First, the agent can be assigned action 3 and paid on inputs. Second, the agent can be offered an outputbased compensation plan and given complete discretion over which action to choose. So far, there is nothing conceptually different from the previous sections. However, there is also an intermediate strategy that can be optimal in which the agent is offered an input-based contract

19 1088 JOURNAL OF POLITICAL ECONOMY and is then given discretion over actions 2 and 3 only: action 1 is excluded. This partial delegation occurs for intermediate levels of uncertainty. To see this, consider the value of a strategy in which the agent is monitored on inputs but can choose from actions 2 and 3. Let the expected value of the first order statistic between these two be given by E[p',21]. This is identical to the problem in example 2 and is given by 1-2J (2x- A) if A< 2x E[p1'2,]- = (6) 0 otherwise. Now consider the value of allowing the agent to also choose the final action 1. Let E[p't,,] refer to the first order statistic chosen from the realization of all three. Straightforward calculations show that E[pl{3,] -E[P1{2] = if ~\2 I-( ](x- -A) 3 A,,ifA<x -- ifa< x 6( x) 2x/ 0 otherwise. (7) The right-hand side of (7) is the return to allowing the agent access to the first action over the restricted choice if he always chooses the action that is preferred by the principal. The problem, of course, is that he may not do so with an input contract. The principal chooses one of these three options: delegate all tasks with output contracts, assign to task 3 with an input contract, or allow choice between 2 and 3 with an input contract. Assigning task 3 to the agent yields profits of A + e* - C(e*) - me + (B/3). Allowing the agent complete discretion with an output-based contract yields profits of E[pl13S] + e* - C(ei) + (B/3) - my. Finally, if the agent is offered an input-based contract and allowed to choose between actions 2 and 3, expected profits are given by B -A + E[pl2] + e* - C(ei*) + -- me This arises as follows. With probability two-thirds, the agent's preferred action is 2 or 3, and he carries out his preferred action. This has a conditional expected value of p, of A/2. With residual probability one-third, action 1 is his preferred action. But this is excluded, and so he chooses the principal's preferred action, yielding an expected equilibrium value of pi of E[p12,1].

20 RISK AND INCENTIVES 1089 It is simple to show that partial delegation will then arise if there exists some value of x (the parameter measuring variance) such that 1 1 B -A + E[P112] + > max{a, E[pl3^] - my- me. (8) Note that for A < x, E[p13}] - E[pn2}1] is increasing in x, as is E[pl2}]. This implies that the value of each strategy can be ordered with respect to the variance of the distribution. As a result, (i) for low variance, the agent is assigned action 3; (ii) for intermediate levels, he may be excluded only from action 1 and offered an input-based contract; and (iii) for high variance, the agent can choose any action but is offered an output-based contract.'6 Thus, for some cases, delegation is assigned in a more continuous way with respect to variance than in the stark outcomes above, yet it remains the case that output contracting is used in sufficiently uncertain settings. C. Communication Note that I have restricted attention to two types of contracts: one in which the worker is delegated the task and one in which the principal simply chooses the set of tasks. Yet there is another option: one in which the agent communicates something to the principal, and the allocation is based on the message sent by the agent. There is one form of communication that can improve the allocation here, in that a Paretoimproving message can be sent: the agent tells the principal the set of actions that are neither (i) the agent's preferred project nor (ii) the principal's preferred project. In the equilibrium in which the principal chooses the action of the agent, excluding these projects benefits both parties.17 Yet the basic insight of the model remains with this extension. Consider the case in which n = 3, and the random variables Pi~ V (0, 16 As a trivial example, consider the case in which m, = oo, so that output-based contracts are never profitable. Then the optimal actions are (i) for low x, assign action 3 to the agent; and (ii) for higher values of x, allow the agent to choose between actions 2 and 3. In both cases, input monitoring is used. 17 It is the indifference of the agent over many actions that allows a role for communication here. If the agent is not indifferent between any activities (in terms of private benefits), there can be no role for communication when output is not observed. But if output is observed, then there is no role for communication since there are no further distortions beyond the monitoring cost.

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